.. currentmodule:: numpy
*************************
Numpy C Code Explanations
*************************
Fanaticism consists of redoubling your efforts when you have forgotten
your aim.
--- *George Santayana*
An authority is a person who can tell you more about something than
you really care to know.
--- *Unknown*
This Chapter attempts to explain the logic behind some of the new
pieces of code. The purpose behind these explanations is to enable
somebody to be able to understand the ideas behind the implementation
somewhat more easily than just staring at the code. Perhaps in this
way, the algorithms can be improved on, borrowed from, and/or
optimized.
Memory model
============
.. index::
pair: ndarray; memory model
One fundamental aspect of the ndarray is that an array is seen as a
"chunk" of memory starting at some location. The interpretation of
this memory depends on the stride information. For each dimension in
an :math:`N` -dimensional array, an integer (stride) dictates how many
bytes must be skipped to get to the next element in that dimension.
Unless you have a single-segment array, this stride information must
be consulted when traversing through an array. It is not difficult to
write code that accepts strides, you just have to use (char \*)
pointers because strides are in units of bytes. Keep in mind also that
strides do not have to be unit-multiples of the element size. Also,
remember that if the number of dimensions of the array is 0 (sometimes
called a rank-0 array), then the strides and dimensions variables are
NULL.
Besides the structural information contained in the strides and
dimensions members of the :ctype:`PyArrayObject`, the flags contain important
information about how the data may be accessed. In particular, the
:cdata:`NPY_ALIGNED` flag is set when the memory is on a suitable boundary
according to the data-type array. Even if you have a contiguous chunk
of memory, you cannot just assume it is safe to dereference a data-
type-specific pointer to an element. Only if the :cdata:`NPY_ALIGNED` flag is
set is this a safe operation (on some platforms it will work but on
others, like Solaris, it will cause a bus error). The :cdata:`NPY_WRITEABLE`
should also be ensured if you plan on writing to the memory area of
the array. It is also possible to obtain a pointer to an unwriteable
memory area. Sometimes, writing to the memory area when the
:cdata:`NPY_WRITEABLE` flag is not set will just be rude. Other times it can
cause program crashes ( *e.g.* a data-area that is a read-only
memory-mapped file).
Data-type encapsulation
=======================
.. index::
single: dtype
The data-type is an important abstraction of the ndarray. Operations
will look to the data-type to provide the key functionality that is
needed to operate on the array. This functionality is provided in the
list of function pointers pointed to by the 'f' member of the
:ctype:`PyArray_Descr` structure. In this way, the number of data-types can be
extended simply by providing a :ctype:`PyArray_Descr` structure with suitable
function pointers in the 'f' member. For built-in types there are some
optimizations that by-pass this mechanism, but the point of the data-
type abstraction is to allow new data-types to be added.
One of the built-in data-types, the void data-type allows for
arbitrary records containing 1 or more fields as elements of the
array. A field is simply another data-type object along with an offset
into the current record. In order to support arbitrarily nested
fields, several recursive implementations of data-type access are
implemented for the void type. A common idiom is to cycle through the
elements of the dictionary and perform a specific operation based on
the data-type object stored at the given offset. These offsets can be
arbitrary numbers. Therefore, the possibility of encountering mis-
aligned data must be recognized and taken into account if necessary.
N-D Iterators
=============
.. index::
single: array iterator
A very common operation in much of NumPy code is the need to iterate
over all the elements of a general, strided, N-dimensional array. This
operation of a general-purpose N-dimensional loop is abstracted in the
notion of an iterator object. To write an N-dimensional loop, you only
have to create an iterator object from an ndarray, work with the
dataptr member of the iterator object structure and call the macro
:cfunc:`PyArray_ITER_NEXT` (it) on the iterator object to move to the next
element. The "next" element is always in C-contiguous order. The macro
works by first special casing the C-contiguous, 1-D, and 2-D cases
which work very simply.
For the general case, the iteration works by keeping track of a list
of coordinate counters in the iterator object. At each iteration, the
last coordinate counter is increased (starting from 0). If this
counter is smaller then one less than the size of the array in that
dimension (a pre-computed and stored value), then the counter is
increased and the dataptr member is increased by the strides in that
dimension and the macro ends. If the end of a dimension is reached,
the counter for the last dimension is reset to zero and the dataptr is
moved back to the beginning of that dimension by subtracting the
strides value times one less than the number of elements in that
dimension (this is also pre-computed and stored in the backstrides
member of the iterator object). In this case, the macro does not end,
but a local dimension counter is decremented so that the next-to-last
dimension replaces the role that the last dimension played and the
previously-described tests are executed again on the next-to-last
dimension. In this way, the dataptr is adjusted appropriately for
arbitrary striding.
The coordinates member of the :ctype:`PyArrayIterObject` structure maintains
the current N-d counter unless the underlying array is C-contiguous in
which case the coordinate counting is by-passed. The index member of
the :ctype:`PyArrayIterObject` keeps track of the current flat index of the
iterator. It is updated by the :cfunc:`PyArray_ITER_NEXT` macro.
Broadcasting
============
.. index::
single: broadcasting
In Numeric, broadcasting was implemented in several lines of code
buried deep in ufuncobject.c. In NumPy, the notion of broadcasting has
been abstracted so that it can be performed in multiple places.
Broadcasting is handled by the function :cfunc:`PyArray_Broadcast`. This
function requires a :ctype:`PyArrayMultiIterObject` (or something that is a
binary equivalent) to be passed in. The :ctype:`PyArrayMultiIterObject` keeps
track of the broadcasted number of dimensions and size in each
dimension along with the total size of the broadcasted result. It also
keeps track of the number of arrays being broadcast and a pointer to
an iterator for each of the arrays being broadcasted.
The :cfunc:`PyArray_Broadcast` function takes the iterators that have already
been defined and uses them to determine the broadcast shape in each
dimension (to create the iterators at the same time that broadcasting
occurs then use the :cfunc:`PyMultiIter_New` function). Then, the iterators are
adjusted so that each iterator thinks it is iterating over an array
with the broadcasted size. This is done by adjusting the iterators
number of dimensions, and the shape in each dimension. This works
because the iterator strides are also adjusted. Broadcasting only
adjusts (or adds) length-1 dimensions. For these dimensions, the
strides variable is simply set to 0 so that the data-pointer for the
iterator over that array doesn't move as the broadcasting operation
operates over the extended dimension.
Broadcasting was always implemented in Numeric using 0-valued strides
for the extended dimensions. It is done in exactly the same way in
NumPy. The big difference is that now the array of strides is kept
track of in a :ctype:`PyArrayIterObject`, the iterators involved in a
broadcasted result are kept track of in a :ctype:`PyArrayMultiIterObject`,
and the :cfunc:`PyArray_BroadCast` call implements the broad-casting rules.
Array Scalars
=============
.. index::
single: array scalars
The array scalars offer a hierarchy of Python types that allow a one-
to-one correspondence between the data-type stored in an array and the
Python-type that is returned when an element is extracted from the
array. An exception to this rule was made with object arrays. Object
arrays are heterogeneous collections of arbitrary Python objects. When
you select an item from an object array, you get back the original
Python object (and not an object array scalar which does exist but is
rarely used for practical purposes).
The array scalars also offer the same methods and attributes as arrays
with the intent that the same code can be used to support arbitrary
dimensions (including 0-dimensions). The array scalars are read-only
(immutable) with the exception of the void scalar which can also be
written to so that record-array field setting works more naturally
(a[0]['f1'] = ``value`` ).
Advanced ("Fancy") Indexing
=============================
.. index::
single: indexing
The implementation of advanced indexing represents some of the most
difficult code to write and explain. In fact, there are two
implementations of advanced indexing. The first works only with 1-D
arrays and is implemented to handle expressions involving a.flat[obj].
The second is general-purpose that works for arrays of "arbitrary
dimension" (up to a fixed maximum). The one-dimensional indexing
approaches were implemented in a rather straightforward fashion, and
so it is the general-purpose indexing code that will be the focus of
this section.
There is a multi-layer approach to indexing because the indexing code
can at times return an array scalar and at other times return an
array. The functions with "_nice" appended to their name do this
special handling while the function without the _nice appendage always
return an array (perhaps a 0-dimensional array). Some special-case
optimizations (the index being an integer scalar, and the index being
a tuple with as many dimensions as the array) are handled in
array_subscript_nice function which is what Python calls when
presented with the code "a[obj]." These optimizations allow fast
single-integer indexing, and also ensure that a 0-dimensional array is
not created only to be discarded as the array scalar is returned
instead. This provides significant speed-up for code that is selecting
many scalars out of an array (such as in a loop). However, it is still
not faster than simply using a list to store standard Python scalars,
because that is optimized by the Python interpreter itself.
After these optimizations, the array_subscript function itself is
called. This function first checks for field selection which occurs
when a string is passed as the indexing object. Then, 0-D arrays are
given special-case consideration. Finally, the code determines whether
or not advanced, or fancy, indexing needs to be performed. If fancy
indexing is not needed, then standard view-based indexing is performed
using code borrowed from Numeric which parses the indexing object and
returns the offset into the data-buffer and the dimensions necessary
to create a new view of the array. The strides are also changed by
multiplying each stride by the step-size requested along the
corresponding dimension.
Fancy-indexing check
--------------------
The fancy_indexing_check routine determines whether or not to use
standard view-based indexing or new copy-based indexing. If the
indexing object is a tuple, then view-based indexing is assumed by
default. Only if the tuple contains an array object or a sequence
object is fancy-indexing assumed. If the indexing object is an array,
then fancy indexing is automatically assumed. If the indexing object
is any other kind of sequence, then fancy-indexing is assumed by
default. This is over-ridden to simple indexing if the sequence
contains any slice, newaxis, or Ellipsis objects, and no arrays or
additional sequences are also contained in the sequence. The purpose
of this is to allow the construction of "slicing" sequences which is a
common technique for building up code that works in arbitrary numbers
of dimensions.
Fancy-indexing implementation
-----------------------------
The concept of indexing was also abstracted using the idea of an
iterator. If fancy indexing is performed, then a :ctype:`PyArrayMapIterObject`
is created. This internal object is not exposed to Python. It is
created in order to handle the fancy-indexing at a high-level. Both
get and set fancy-indexing operations are implemented using this
object. Fancy indexing is abstracted into three separate operations:
(1) creating the :ctype:`PyArrayMapIterObject` from the indexing object, (2)
binding the :ctype:`PyArrayMapIterObject` to the array being indexed, and (3)
getting (or setting) the items determined by the indexing object.
There is an optimization implemented so that the :ctype:`PyArrayIterObject`
(which has it's own less complicated fancy-indexing) is used for
indexing when possible.
Creating the mapping object
^^^^^^^^^^^^^^^^^^^^^^^^^^^
The first step is to convert the indexing objects into a standard form
where iterators are created for all of the index array inputs and all
Boolean arrays are converted to equivalent integer index arrays (as if
nonzero(arr) had been called). Finally, all integer arrays are
replaced with the integer 0 in the indexing object and all of the
index-array iterators are "broadcast" to the same shape.
Binding the mapping object
^^^^^^^^^^^^^^^^^^^^^^^^^^
When the mapping object is created it does not know which array it
will be used with so once the index iterators are constructed during
mapping-object creation, the next step is to associate these iterators
with a particular ndarray. This process interprets any ellipsis and
slice objects so that the index arrays are associated with the
appropriate axis (the axis indicated by the iteraxis entry
corresponding to the iterator for the integer index array). This
information is then used to check the indices to be sure they are
within range of the shape of the array being indexed. The presence of
ellipsis and/or slice objects implies a sub-space iteration that is
accomplished by extracting a sub-space view of the array (using the
index object resulting from replacing all the integer index arrays
with 0) and storing the information about where this sub-space starts
in the mapping object. This is used later during mapping-object
iteration to select the correct elements from the underlying array.
Getting (or Setting)
^^^^^^^^^^^^^^^^^^^^
After the mapping object is successfully bound to a particular array,
the mapping object contains the shape of the resulting item as well as
iterator objects that will walk through the currently-bound array and
either get or set its elements as needed. The walk is implemented
using the :cfunc:`PyArray_MapIterNext` function. This function sets the
coordinates of an iterator object into the current array to be the
next coordinate location indicated by all of the indexing-object
iterators while adjusting, if necessary, for the presence of a sub-
space. The result of this function is that the dataptr member of the
mapping object structure is pointed to the next position in the array
that needs to be copied out or set to some value.
When advanced indexing is used to extract an array, an iterator for
the new array is constructed and advanced in phase with the mapping
object iterator. When advanced indexing is used to place values in an
array, a special "broadcasted" iterator is constructed from the object
being placed into the array so that it will only work if the values
used for setting have a shape that is "broadcastable" to the shape
implied by the indexing object.
Universal Functions
===================
.. index::
single: ufunc
Universal functions are callable objects that take :math:`N` inputs
and produce :math:`M` outputs by wrapping basic 1-D loops that work
element-by-element into full easy-to use functions that seamlessly
implement broadcasting, type-checking and buffered coercion, and
output-argument handling. New universal functions are normally created
in C, although there is a mechanism for creating ufuncs from Python
functions (:func:`frompyfunc`). The user must supply a 1-D loop that
implements the basic function taking the input scalar values and
placing the resulting scalars into the appropriate output slots as
explaine n implementation.
Setup
-----
Every ufunc calculation involves some overhead related to setting up
the calculation. The practical significance of this overhead is that
even though the actual calculation of the ufunc is very fast, you will
be able to write array and type-specific code that will work faster
for small arrays than the ufunc. In particular, using ufuncs to
perform many calculations on 0-D arrays will be slower than other
Python-based solutions (the silently-imported scalarmath module exists
precisely to give array scalars the look-and-feel of ufunc-based
calculations with significantly reduced overhead).
When a ufunc is called, many things must be done. The information
collected from these setup operations is stored in a loop-object. This
loop object is a C-structure (that could become a Python object but is
not initialized as such because it is only used internally). This loop
object has the layout needed to be used with PyArray_Broadcast so that
the broadcasting can be handled in the same way as it is handled in
other sections of code.
The first thing done is to look-up in the thread-specific global
dictionary the current values for the buffer-size, the error mask, and
the associated error object. The state of the error mask controls what
happens when an error-condiction is found. It should be noted that
checking of the hardware error flags is only performed after each 1-D
loop is executed. This means that if the input and output arrays are
contiguous and of the correct type so that a single 1-D loop is
performed, then the flags may not be checked until all elements of the
array have been calcluated. Looking up these values in a thread-
specific dictionary takes time which is easily ignored for all but
very small arrays.
After checking, the thread-specific global variables, the inputs are
evaluated to determine how the ufunc should proceed and the input and
output arrays are constructed if necessary. Any inputs which are not
arrays are converted to arrays (using context if necessary). Which of
the inputs are scalars (and therefore converted to 0-D arrays) is
noted.
Next, an appropriate 1-D loop is selected from the 1-D loops available
to the ufunc based on the input array types. This 1-D loop is selected
by trying to match the signature of the data-types of the inputs
against the available signatures. The signatures corresponding to
built-in types are stored in the types member of the ufunc structure.
The signatures corresponding to user-defined types are stored in a
linked-list of function-information with the head element stored as a
``CObject`` in the userloops dictionary keyed by the data-type number
(the first user-defined type in the argument list is used as the key).
The signatures are searched until a signature is found to which the
input arrays can all be cast safely (ignoring any scalar arguments
which are not allowed to determine the type of the result). The
implication of this search procedure is that "lesser types" should be
placed below "larger types" when the signatures are stored. If no 1-D
loop is found, then an error is reported. Otherwise, the argument_list
is updated with the stored signature --- in case casting is necessary
and to fix the output types assumed by the 1-D loop.
If the ufunc has 2 inputs and 1 output and the second input is an
Object array then a special-case check is performed so that
NotImplemented is returned if the second input is not an ndarray, has
the __array_priority\__ attribute, and has an __r{op}\__ special
method. In this way, Python is signaled to give the other object a
chance to complete the operation instead of using generic object-array
calculations. This allows (for example) sparse matrices to override
the multiplication operator 1-D loop.
For input arrays that are smaller than the specified buffer size,
copies are made of all non-contiguous, mis-aligned, or out-of-
byteorder arrays to ensure that for small arrays, a single-loop is
used. Then, array iterators are created for all the input arrays and
the resulting collection of iterators is broadcast to a single shape.
The output arguments (if any) are then processed and any missing
return arrays are constructed. If any provided output array doesn't
have the correct type (or is mis-aligned) and is smaller than the
buffer size, then a new output array is constructed with the special
UPDATEIFCOPY flag set so that when it is DECREF'd on completion of the
function, it's contents will be copied back into the output array.
Iterators for the output arguments are then processed.
Finally, the decision is made about how to execute the looping
mechanism to ensure that all elements of the input arrays are combined
to produce the output arrays of the correct type. The options for loop
execution are one-loop (for contiguous, aligned, and correct data-
type), strided-loop (for non-contiguous but still aligned and correct
data-type), and a buffered loop (for mis-aligned or incorrect data-
type situations). Depending on which execution method is called for,
the loop is then setup and computed.
Function call
-------------
This section describes how the basic universal function computation
loop is setup and executed for each of the three different kinds of
execution possibilities. If :cdata:`NPY_ALLOW_THREADS` is defined during
compilation, then the Python Global Interpreter Lock (GIL) is released
prior to calling all of these loops (as long as they don't involve
object arrays). It is re-acquired if necessary to handle error
conditions. The hardware error flags are checked only after the 1-D
loop is calcluated.
One Loop
^^^^^^^^
This is the simplest case of all. The ufunc is executed by calling the
underlying 1-D loop exactly once. This is possible only when we have
aligned data of the correct type (including byte-order) for both input
and output and all arrays have uniform strides (either contiguous,
0-D, or 1-D). In this case, the 1-D computational loop is called once
to compute the calculation for the entire array. Note that the
hardware error flags are only checked after the entire calculation is
complete.
Strided Loop
^^^^^^^^^^^^
When the input and output arrays are aligned and of the correct type,
but the striding is not uniform (non-contiguous and 2-D or larger),
then a second looping structure is employed for the calculation. This
approach converts all of the iterators for the input and output
arguments to iterate over all but the largest dimension. The inner
loop is then handled by the underlying 1-D computational loop. The
outer loop is a standard iterator loop on the converted iterators. The
hardware error flags are checked after each 1-D loop is completed.
Buffered Loop
^^^^^^^^^^^^^
This is the code that handles the situation whenever the input and/or
output arrays are either misaligned or of the wrong data-type
(including being byte-swapped) from what the underlying 1-D loop
expects. The arrays are also assumed to be non-contiguous. The code
works very much like the strided loop except for the inner 1-D loop is
modified so that pre-processing is performed on the inputs and post-
processing is performed on the outputs in bufsize chunks (where
bufsize is a user-settable parameter). The underlying 1-D
computational loop is called on data that is copied over (if it needs
to be). The setup code and the loop code is considerably more
complicated in this case because it has to handle:
- memory allocation of the temporary buffers
- deciding whether or not to use buffers on the input and output data
(mis-aligned and/or wrong data-type)
- copying and possibly casting data for any inputs or outputs for which
buffers are necessary.
- special-casing Object arrays so that reference counts are properly
handled when copies and/or casts are necessary.
- breaking up the inner 1-D loop into bufsize chunks (with a possible
remainder).
Again, the hardware error flags are checked at the end of each 1-D
loop.
Final output manipulation
-------------------------
Ufuncs allow other array-like classes to be passed seamlessly through
the interface in that inputs of a particular class will induce the
outputs to be of that same class. The mechanism by which this works is
the following. If any of the inputs are not ndarrays and define the
:obj:`__array_wrap__` method, then the class with the largest
:obj:`__array_priority__` attribute determines the type of all the
outputs (with the exception of any output arrays passed in). The
:obj:`__array_wrap__` method of the input array will be called with the
ndarray being returned from the ufunc as it's input. There are two
calling styles of the :obj:`__array_wrap__` function supported. The first
takes the ndarray as the first argument and a tuple of "context" as
the second argument. The context is (ufunc, arguments, output argument
number). This is the first call tried. If a TypeError occurs, then the
function is called with just the ndarray as the first argument.
Methods
-------
Their are three methods of ufuncs that require calculation similar to
the general-purpose ufuncs. These are reduce, accumulate, and
reduceat. Each of these methods requires a setup command followed by a
loop. There are four loop styles possible for the methods
corresponding to no-elements, one-element, strided-loop, and buffered-
loop. These are the same basic loop styles as implemented for the
general purpose function call except for the no-element and one-
element cases which are special-cases occurring when the input array
objects have 0 and 1 elements respectively.
Setup
^^^^^
The setup function for all three methods is ``construct_reduce``.
This function creates a reducing loop object and fills it with
parameters needed to complete the loop. All of the methods only work
on ufuncs that take 2-inputs and return 1 output. Therefore, the
underlying 1-D loop is selected assuming a signature of [ ``otype``,
``otype``, ``otype`` ] where ``otype`` is the requested reduction
data-type. The buffer size and error handling is then retrieved from
(per-thread) global storage. For small arrays that are mis-aligned or
have incorrect data-type, a copy is made so that the un-buffered
section of code is used. Then, the looping strategy is selected. If
there is 1 element or 0 elements in the array, then a simple looping
method is selected. If the array is not mis-aligned and has the
correct data-type, then strided looping is selected. Otherwise,
buffered looping must be performed. Looping parameters are then
established, and the return array is constructed. The output array is
of a different shape depending on whether the method is reduce,
accumulate, or reduceat. If an output array is already provided, then
it's shape is checked. If the output array is not C-contiguous,
aligned, and of the correct data type, then a temporary copy is made
with the UPDATEIFCOPY flag set. In this way, the methods will be able
to work with a well-behaved output array but the result will be copied
back into the true output array when the method computation is
complete. Finally, iterators are set up to loop over the correct axis
(depending on the value of axis provided to the method) and the setup
routine returns to the actual computation routine.
Reduce
^^^^^^
.. index::
triple: ufunc; methods; reduce
All of the ufunc methods use the same underlying 1-D computational
loops with input and output arguments adjusted so that the appropriate
reduction takes place. For example, the key to the functioning of
reduce is that the 1-D loop is called with the output and the second
input pointing to the same position in memory and both having a step-
size of 0. The first input is pointing to the input array with a step-
size given by the appropriate stride for the selected axis. In this
way, the operation performed is
.. math::
:nowrap:
\begin{align*}
o & = & i[0] \\
o & = & i[k]\textrm{}o\quad k=1\ldots N
\end{align*}
where :math:`N+1` is the number of elements in the input, :math:`i`,
:math:`o` is the output, and :math:`i[k]` is the
:math:`k^{\textrm{th}}` element of :math:`i` along the selected axis.
This basic operations is repeated for arrays with greater than 1
dimension so that the reduction takes place for every 1-D sub-array
along the selected axis. An iterator with the selected dimension
removed handles this looping.
For buffered loops, care must be taken to copy and cast data before
the loop function is called because the underlying loop expects
aligned data of the correct data-type (including byte-order). The
buffered loop must handle this copying and casting prior to calling
the loop function on chunks no greater than the user-specified
bufsize.
Accumulate
^^^^^^^^^^
.. index::
triple: ufunc; methods; accumulate
The accumulate function is very similar to the reduce function in that
the output and the second input both point to the output. The
difference is that the second input points to memory one stride behind
the current output pointer. Thus, the operation performed is
.. math::
:nowrap:
\begin{align*}
o[0] & = & i[0] \\
o[k] & = & i[k]\textrm{}o[k-1]\quad k=1\ldots N.
\end{align*}
The output has the same shape as the input and each 1-D loop operates
over :math:`N` elements when the shape in the selected axis is :math:`N+1`.
Again, buffered loops take care to copy and cast the data before
calling the underlying 1-D computational loop.
Reduceat
^^^^^^^^
.. index::
triple: ufunc; methods; reduceat
single: ufunc
The reduceat function is a generalization of both the reduce and
accumulate functions. It implements a reduce over ranges of the input
array specified by indices. The extra indices argument is checked to
be sure that every input is not too large for the input array along
the selected dimension before the loop calculations take place. The
loop implementation is handled using code that is very similar to the
reduce code repeated as many times as there are elements in the
indices input. In particular: the first input pointer passed to the
underlying 1-D computational loop points to the input array at the
correct location indicated by the index array. In addition, the output
pointer and the second input pointer passed to the underlying 1-D loop
point to the same position in memory. The size of the 1-D
computational loop is fixed to be the difference between the current
index and the next index (when the current index is the last index,
then the next index is assumed to be the length of the array along the
selected dimension). In this way, the 1-D loop will implement a reduce
over the specified indices.
Mis-aligned or a loop data-type that does not match the input and/or
output data-type is handled using buffered code where-in data is
copied to a temporary buffer and cast to the correct data-type if
necessary prior to calling the underlying 1-D function. The temporary
buffers are created in (element) sizes no bigger than the user
settable buffer-size value. Thus, the loop must be flexible enough to
call the underlying 1-D computational loop enough times to complete
the total calculation in chunks no bigger than the buffer-size.